We present experiments studying the coherent motion of atoms in crystals made from on and off resonant light. The experiments confirm that inside the light-field atoms fulfilling the Bragg condition form a standing matter wave pattern. As a consequence we observed anomalous transmission of atoms through resonant light fields.

Tailored complex potentials for atoms can be made of two overlapping standing light waves, one on resonance and one far detuned. The observed diffraction asymmetry of Bragg diffraction of such light structures is due to a corresponding asymmetry of the Fourier components of the potential. In crystal physics this is known as a violation of Friedel's law. [S0031-9007(97) PACS numbers: 61.12.Bt It is often found that concepts of photon optics can be adapted to matter-wave optics. In our article we choose the conjugate approach. We use the simplicity of the interaction between light and matter waves to design complex periodic potentials for the matter waves and reveal optical concepts. As an example, we investigate a violation of Friedel's law due to fundamental optical principles in a very controlled system.Typically, diffraction phenomena are invariant under an inversion of the crystal, even when the elementary cell of the crystal possesses no symmetry. This empirical rule is generally referred to as Friedel's law [1]. However, violations of this rule are known from diffraction experiments of x rays or electrons at solid state crystals [2], for example, due to the presence of "anomalous" (absorptive) scatterers. In this Letter we present a violation of Friedel's law in a very different system, where atomic matter waves are diffracted at specially designed "crystals" of light [3,4]. The diffraction asymmetry is due to the interaction of both "normal" and anomalous scattering at superposed refractive and absorptive subcrystals, respectively [5]. This mechanism even works although our light crystal obviously cannot be really absorptive for the atoms, but only changes their internal state.In our experiment ( Fig. 1), we detect atom intensities depending on the atom's incidence angle at the light crystals, and their diffraction angle. In the case of spatial coincidence between the refractive and the absorptive parts of the crystals we obtain symmetric diffraction, as shown in Fig. 1(b). This corresponds to the normal situation in solid state crystals, where Friedel's law is obeyed. However, a violation of Friedel's law is demonstrated in Figs. 1(a) and 1(c), where Bragg diffraction is dominant in one direction. There, the absorptive and refractive index parts of the crystals are arranged such that they are out of phase by 6p͞2.In the remainder of this Letter, we will show that this diffraction asymmetry can be understood by evaluating the Fourier composition of the resulting complex potential, and employing dynamical diffraction theory. This means specifically that the effect of the total crystal potential cannot be separated into the individual actions of its components. However, in the weak diffraction limit a more intuitive picture can be justified, which is presented in the following.The asymmetry can be understood as an interference effect between diffraction at refractive and absorptive "subcrystals" spatially displaced with respect to each other (Fig. 2). Generally, there is a p͞2 phase shift even between wa...

We build amplitude, i.e., absorptive masks, for neutral atoms using light, reversing the roles of light and atoms as compared to conventional optics. These masks can be used both to create and to probe spatially well-defined atomic distributions. The resolution of these masks can be significantly better than the optical wavelength. Applications range from atom lithography to fundamental atom optical experiments.

Temporal light modulation methods which are of great practical importance in optical technology, are emulated with matter waves. This includes generation and tailoring of matter-wave sidebands, using amplitude and phase modulation of an atomic beam. In the experiments atoms are Bragg diffracted at standing light fields, which are periodically modulated in intensity or frequency. This gives rise to a generalized Bragg situation under which the atomic matter waves are both diffracted and coherently shifted in their de Broglie frequency. In particular, we demonstrate creation of complex and non-Hermitian matter-wave modulations. One interesting case is a potential with a time-dependent complex helicity ͓Vϰexp(it)͔, which produces a purely lopsided energy transfer between the atoms and the photons, and thus violates the usual symmetry between absorption and stimulated emission of energy quanta. Possible applications range from atom cooling over advanced atomic interferometers to a new type of mass spectrometer.

Atoms in light crystals formed by a standing light wave are a model system to study the propagation of matter waves in periodic potentials. The encountered phenomena can be described by dynamical diffraction theory which has been extensively studied for x-ray, electron, and neutron scattering from solid state crystals. In this paper we show that an atomic de Broglie wave traversing a standing light wave allows investigation of predictions of dynamical diffraction theory which were previously experimentally not accessible. We present standard diffraction efficiency characterizations for pure absorptive and pure refractive crystals. Additionally we were able to measure directly the total atomic wave field formed inside a refractive and an absorptive crystal and to confirm the predicted absolute position of the atomic wave field with respect to the lattice planes. By superposing two standing light waves of different frequency we tailored a new kind of crystal potential where Friedel's law about usual diffraction symmetries is maximally violated. ͓S1050-2947͑99͒02207-6͔

We demonstrate how Bragg diffraction of atomic matter waves at a time-modulated thick standing light wave can be used to coherently shift the de Broglie frequency of the diffracted atoms. The coherent frequency shift is experimentally confirmed by interferometric superposition of modulated and unmodulated atoms resulting in time-dependent interference fringes. Our frequency shifter for atomic matter waves is a generalization of an acousto-optic frequency shifter for photons.[S0031-9007 (96)01871-6] PACS numbers: 03.75.Be, 03.75.Dg, 32.80.Pj Investigations of time-dependent matter wave phenomena are a current field of research in order to study fundamental predictions of quantum theory. In neutron optics such experiments have been performed in recent years. Neutrons were deflected by time-dependent mirrors [1,2], slowed by diffraction at moving gratings and mirrors [3], or sent through time-dependent potentials [4]. Recently, experiments have been performed with atomic matter waves [5,6] where atoms released from a magnetooptical trap were reflected by a vibrating mirror. Many frequency sidebands of the reflected de Broglie waves have been detected in these experiments.In the present Letter we demonstrate an easy modulation technique for a continuous atomic beam, where, in contrast to [5,6], the interaction time between the atoms and the light field is much larger than the temporal modulation period. Temporal diffraction in this regime can be viewed as the time equivalent of spatial Bragg scattering.We start by comparing our time-modulated scattering experiments with standard Bragg diffraction [7-10]. An important feature of Bragg diffraction is its angle and velocity selectivity, and the appearance of only one diffracted beam, in contrast to diffraction at thin gratings [11,12] (Raman-Nath regime). Atoms are only diffracted if their velocity is matched to their incidence angle at the standing light wave, the "light crystal," by the Bragg condition; otherwise they just traverse the light field without deflection. This is demonstrated in the Bragg diffraction experiments plotted in Fig. 1. In case (A) the incidence angle of the atomic beam is the static Bragg angle (17.7 mrad). Two peaks result, corresponding to diffracted (arrow) and transmitted atoms, respectively. (B) shows the same experiment for atoms incident at an angle which differs by 44.2 mrad (approximately 2.5 Bragg angles) from the stationary Bragg angle. Only the peak of transmitted atoms is obtained.A new observation is presented in (C). Here, the atoms are incident at the same detuned angle as in (B), but additionally the intensity of the standing light wave is modulated with a specific frequency (75 kHz) which is matched to the detuning of the incidence angle as explained below. Now, diffracted atoms are observed again (arrow) at the same position as in the case (A) of static diffraction. Thus, a particular intensity modulation frequency of the standing light wave compensates a detuning of the Bragg angle and retrieves the conditions enabling diffr...

Bragg diffraction of atoms at thick standing light waves requires that the wavematching condition is fulfilled. This usually means that the atomic beam crosses the light wave exactly at the Bragg angle. Nevertheless, our experiments also demonstrate Bragg diffraction at detuned angles if the amplitude of the standing light wave is temporally modulated with an appropriate frequency. If, on the other hand, the phase of the light wave is modulated no diffraction is observed. Both modulation processes produce frequency sidebands which set up 'slowly travelling standing waves' in front of a retro-reflection mirror. Atoms are diffracted at these 'almost standing' light waves in a similar way to photons at the travelling sound waves in an acousto-optic modulator. The frequency of the diffracted de Broglie waves is assumed to be shifted by the intensity modulation frequency. The different results using amplitude-and phase-modulated light waves are due to interference between the diffraction contributions of the individual frequency sidebands contained in the standing light wave.

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